where \(\mathbf {Y}_\ell \in \mathbb {R}^{n_\ell \times L_\ell }\)  is the observed low spatial-high spectral resolution image with \(L_\ell \)  bands and \(n_\ell \)  pixels, \(\mathbf {Y}_h \in \mathbb {R}^{n_h\times L_h}\)  is the observed high spatial-low spectral resolution image with \(L_h\)  bands, and \(\mathbf {Z}\in \mathbb {R}^{n_h \times L_\ell }\)  is the ground-truth image having high spatial and spectral resolutions. Typically, \(L_h \ll L_\ell \)  and \(n_\ell \ll n_h\) . In this work, we assume that the observed images \(\mathbf {Y}_\ell \)  and \(\mathbf {Y}_h\)  are preprocessed and aligned [1]}, [2]}, [3]}. The intrinsic parameters in (REF ) are the blur operator \(\mathbf {B}\in \mathbb {R}^{n_h \times n_h}\) , the decimation operator \(\mathbf {S}\in \mathbb {R}^{n_\ell \times n_h}\) , and the spectral response operator \(\mathbf {R}\in \mathbb {R}^{L_\ell \times L_h }\)  [4]}, [5]}; we assume that these parameters are known. \(\mathbf {N}_\ell \)  and \(\mathbf {N}_h\)  are white Gaussian noise.
Though the variable \(\mathbf {Z}\)  lies in a high-dimensional space (\(n_h L_\ell \)  unknowns), it has a low rank due to the correlation between different bands. This observation is used to cut down the number of variables via the low-rank representation \(\mathbf {Z}=\mathbf {X}\mathbf {E}\) , where \(\mathbf {E}\in \mathbb {R}^{L_s \times L_\ell }\)  is a fixed matrix (obtained using SVD or vertex component analysis [6]} of \(\mathbf {Y}_\ell \) ) and \(\mathbf {X}\in \mathbb {R}^{n_h \times L_s}\)  is the low-dimensional variable having fewer unknowns [7]}, [8]}, [3]}.
Combining this with (REF ), the image fusion task can be posed as a variational problem [7]}:
\(\min _{\mathbf {X}} \ \ \ell (\mathbf {X}) + \lambda _2 \phi (\mathbf {X}),\) 
